This semester, I am assisting my advisor with his honors calculus class. As he is out of town today, I will be giving the lecture. We decided to show off a rather beautiful little bit of mathematics which makes use of many of the tools that we have developed since September, including an interesting application of Riemann sums. We are going to be trying to evaluate \(\newcommand{\8}{\infty}\lim_{n\to\8} \frac{n}{(n!)^{1/n}}\).
For the last several months, I have been working hard on getting my masters thesis written. This is a somewhat tedious activity—the research phase is essentially done, and now I have to get everything typed up all nicely. Other interests have taken something of a backseat. One of the things that I have managed to do in order to stay sane is make a lot of cookies. I have been experimenting with recipes, and came up with something pretty good this week.
When cooking, we need three things: a collection of ingredients (say butter, sugar, flour, &c.), a description of a technique for combining them (a recipe), and some tools for actually getting the work done (bowls and knives, as well as some level of skill in their use). Mathematics is much the same—in order to practice mathematics, we need ingredients and techniques for combining them. In a very broad sense, the basic ingredients are the axioms, which can be mixed together using first-order logic and an ability to use these to produce theorems.
Here’s an interesting little problem that came across my desk this afternoon: how much time is \(10!\) seconds? Is it a duration that is best measured in seconds? days? centuries? And, perhaps more importantly, what is the best way of figuring it out? Think about it for a minute, then check below the fold for the answer, which is a surprisingly round number!
The question of solvability is one of the driving questions in mathematics. That is, given an equation, can it be solved? and if so, what are the solutions?
There are some equations that are very easily solved. For instance, the linear equation \(ax + b = 0\) can easily be solved by subtracting \(b\) from each side then dividing through by \(a\). That is, \(x=-b/a\) is a solution to any equation of the form \(ax+b=0\). Quadratic equations, i.e. those of the form \(ax^2+bx+c=0\), also have solutions. In fact, they generally have two solutions that can be found using the quadratic formula, and which look like
\[ \frac{-b\pm\sqrt{b^2-4ac}}{2a}.\]
There are also “nice” formulae for solving cubic and quartic (degree three and degree four polynomial) equations. These solutions involve only elementary mathematical operations like addition, multiplication, and taking roots. So what about higher order polynomials? For instance, does the equation
\[ ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0 \]
have any solutions, and, if so, what are those solutions?
In the vein of the silliness suggested by Patrick Vennebush at Math Jokes 4 Mathy Folks, I present Gottfried Leibnitz, jeans model. Gauss, eat your heart out.
For those that care, here is the logo:
Clearly, I have too much time on my hands. Back to grading.
